We can factor out 100 (i.e., 10000 under the square root) since it does not change the maximum point. So let . Then . From here, iff . Taking square of both sides and adding like terms, we get , i.e., . One must also check that both roots exist and are not zero for this x because by eliminating the denominators and taking squares we may have added spurious roots.

You can use WolframAlpha to solve this equation like this. While it is not recommended to use it to avoid solving things by hand, it can be used to verify that each of your manual steps.